Exact results for optimal phenotypic switching rates

Exact results for optimal phenotypic switching

rates A Jamie Wood, Bernadett Gaal, Jon Pitchford

Phenotypic switching

Phenotypic switching

Importance

May be very relevant to infection, the cull is the antibiotic, regrowth of nasty phenotype still occurs. NOT a mutation.

When to switch? When not to switch? How has switching evolved? What is optimal switching?

Previous Work

Reluga 2005Salathé et al. 2009

Kussell and Leibler, 2005

Ishii et al. 1989

Thattai and van Oudenaarden, 2004

Lachmann and Jablonka, 1996

...optimal rate of switching is approximately equal to the rate of environmental change.

...environment that periodically fluctuates between two states

...what about when environments are different?

Problem setup...

Periodic environments, “A” and “B”. We can think of one of the environments as a period of antimicrobial treatment

AEA AE

BBE

A BEB

Problem setup, in maths

Long term max growthrate:

where

kk

kkM

AB

AA

EA

kk

kkM

BB

BA

EB

BAAE

AA n kn kn

dt

dn

Ank nk ndt

dnBB

EB

B x

x EMdt

d

x(0)e...eeeeex(t)

timesq

TMTMTMTMTMTM BBEAAEBBEAAEBBEAAE

BA

TMTM

TT

)e(elnr

BBEAAE

The Solution...

where

))(4k(4k),,T,T(k,

),,T,T(k,4k4k

e

2B

22A

2BABA

2

BABA2B

22A

2

)T(T2

1k))(T(T BBAABA

(k,TA,TB,A,B) (4k 2 AB)sAsB 4k2 A2 4k2 B

2cAcB

and

si sinh(Ti

24k2 i

2 ),c i cosh(Ti

24k2 i

2 )

Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

Some definitions

Universal switching rateBA E

BE

A k

Fitness of fitter types is the same

BEA

AEB

Differing fitness penalties in the two different environments

AT Time spent in environment ATime spent in environment B

BTGaal, Pitchford, Wood. Genetics Vol. 184, 1113

Key result – not switching is good

10T50T 20T

Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

Key result – discontinuous change

r

k

0

Max

Max

5.0

Max

0.1

Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

More exotic outcomes

Allowing time periods to also vary

Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

A few other ideas

Mathematical framework is well established – can we move to more interesting examples for biology?Fitness? Is switching directly or indirectly affecting fitness? Spatial position in a biofilm for instance. What implications do these results have for evolution of switching?

Thanks to:

Bernadett GaalJon Pitchford Marjan van der Woude